A lumped parameter model for the acoustic power output from a vibrating structure

Previous applications of lumped parameter models to acoustic radiation problems assume that the characteristic dimension of the vibrating structure is small in comparison to the acoustic wavelength. In this paper, the frequency range of the lumped parameter model is extended by dividing the surface of the structure into elements and characterizing the amplitude of the radiation from each element by its volume velocity. The model is derived by truncating all but the lowest-order (monopole) terms of a multipole expansion for the acoustic power output. The multipole expansion differs from those derived previously because it is based on elemental quantities rather than global quantities. By comparing the full multipole expansion for the power output to the lumped parameter model, the error in the lumped parameter model as a function of the acoustic and structural wavelengths (k and K) and the size of the largest surface element (L) is determined. This approach is general and provides a means of determining bounds on the accuracy of any lumped parameter model based on elemental quantities. For example, the analysis predicts that when the overall volume velocity of a vibrating structure is nonzero, the maximum possible error in the lumped parameter model is equal to C(kL)(KL), where C is a constant. Likewise, when the overall volume velocity of a vibrating structure is zero, the model predicts that the maximum possible error in the lumped parameter model is equal to C'(KL)(L/R₁₂), where C' is another constant, and R₁₂ is the largest distance between any two points on the structure. The results of the analysis show that it is desirable to formulate acoustic models in terms of elemental volume velocities, because the power output predicted by any such model converges absolutely to the correct solution as the element mesh is refined.